I'm trying to prove the next result on differential equations.
Let $H:\mathbb{R}^n\to\mathbb{R}^n$ be of class $C^1$ and $f(t,x)$ a continuous function on $\mathbb{R}\times\mathbb{R}^n$ with $f(t,H(x))=DH(x)\cdot f(t,x)$ (and $D$ is the differential operator). If $f$ is Lipschitz and $\phi(t,t_0,x_0)$ is a solution of $x'=f(x), x(t_0)=x_0$, then $$\phi(t,t_0,H(x))=H(\phi(t,t_0,x_0))$$
Here is my attempt:
\begin{eqnarray} D[H(\phi(t,t_0,x_0))]&=&DH(x)\cdot \phi'(t,t_0,x_0)\\ &=&DH(x)\cdot f(t,x)\\ &=& f(t,H(x))\\ &=&D(\phi(t,t_0,H(x)) \end{eqnarray} Is this correct? And, does this (together with the lipschitz condition) imply the conclusion?